Gaussian graphical models assume that the random vector \({x} = [x_1, \ldots , x_p]^\top \)  is centered multivariate Gaussian with covariance matrix \(\mathbb {E}[{x}{x}^\top ] = {\Sigma }\) , denoted \({x}\sim \mathcal {N}({0},{\Sigma })\) .
For the corresponding Gaussian Markov random field, an undirected graph is matched to the variables as follows: each variable corresponds to a vertex, and an edge is present between two vertices if the corresponding random variables are conditionally dependent given the others [1]}, [2]}.
The support of the precision matrix \({\Theta } = {\Sigma }^{-1}\)  directly accounts for this conditional dependency, since
\({\rm corr}\left[ x_q x_\ell | {x}_{[\![ 1,p ]\!]\backslash \lbrace q,\ell \rbrace } \right]= - {\Theta }_{q\ell } / \sqrt{{\Theta }_{qq}{\Theta }_{\ell \ell }}.\) 
